# Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds

arXiv: https://arxiv.org/abs/2403.06560 · HTML: https://mlsanity.com/paper/2403.06560/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Mar 2024.

## Built on by (most cited first, 28 of 28)

- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)
- [Lightspeed Geometric Dataset Distance via Sliced Optimal Transport](https://mlsanity.com/paper/2501.18901/) (2025)
- [A Novel Sliced Fused Gromov-Wasserstein Distance](https://mlsanity.com/paper/2508.02364/) (2025)
- [A User's Guide to Sampling Strategies for Sliced Optimal Transport](https://mlsanity.com/paper/2502.02275/) (2025)
- [Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions](https://mlsanity.com/paper/2404.15378/) (2024)
- [Fast Estimation of Wasserstein Distances via Regression on Sliced Wasserstein Distances](https://mlsanity.com/paper/2509.20508/) (2025)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Hyperbolic Busemann Neural Networks](https://mlsanity.com/paper/2602.18858/) (2026)
- [Summarizing Bayesian Nonparametric Mixture Posterior -- Sliced Optimal Transport Metrics for Gaussian Mixtures](https://mlsanity.com/paper/2411.14674/) (2024)
- [DDEQs: Distributional Deep Equilibrium Models through Wasserstein Gradient Flows](https://mlsanity.com/paper/2503.01140/) (2025)
- [Understanding Learning with Sliced-Wasserstein Requires Rethinking Informative Slices](https://mlsanity.com/paper/2411.10651/) (2024)
- [Amortized Optimal Transport from Sliced Potentials](https://mlsanity.com/paper/2604.15114/) (2026)
- [Sliced-Regularized Optimal Transport](https://mlsanity.com/paper/2604.23944/) (2026)
- [Gaussians on their Way: Wasserstein-Constrained 4D Gaussian Splatting with State-Space Modeling](https://mlsanity.com/paper/2412.00333/) (2024)
- [Repulsive Monte Carlo on the sphere for the sliced Wasserstein distance](https://mlsanity.com/paper/2509.10166/) (2025)
- [Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing](https://mlsanity.com/paper/2510.04579/) (2025)
- [Sliced Inner Product Gromov-Wasserstein Distances](https://mlsanity.com/paper/2605.08546/) (2026)
- [Streaming Sliced Optimal Transport](https://mlsanity.com/paper/2505.06835/) (2025)
- [ReSWD: ReSTIR'd, not shaken. Combining Reservoir Sampling and Sliced Wasserstein Distance for Variance Reduction](https://mlsanity.com/paper/2510.01061/) (2025)
- [Revisiting the Sliced Wasserstein Kernel for persistence diagrams: a Figalli-Gigli approach](https://mlsanity.com/paper/2602.06539/) (2026)
- [Reliable Modeling of Distribution Shifts via Displacement-Reshaped Optimal Transport](https://mlsanity.com/paper/2605.04965/) (2026)
- [ASAP: Amortized Doubly-Stochastic Attention via Sliced Dual Projection](https://mlsanity.com/paper/2605.12879/) (2026)
- [A Sliced-Wasserstein Framework on Correlation Matrices for EEG Decoding](https://mlsanity.com/paper/2606.06104/) (2026)
- [Conditional Random Ordered Transport Spaces](https://mlsanity.com/paper/2606.08113/) (2026)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)
- [1-Lipschitz Neural Networks on Hadamard Manifolds](https://mlsanity.com/paper/2607.19335/) (2026)
- [Anatomy-Informed Neural Networks: Encoding Anatomic Priors in Loss and Architecture, with an SE(3) Formulation of Guidewire-Induced Aortoiliac Deformation](https://mlsanity.com/paper/2608.21332/) (2026)

## Abstract (arXiv)

While many Machine Learning methods were developed or transposed on Riemannian manifolds to tackle data with known non Euclidean geometry, Optimal Transport (OT) methods on such spaces have not received much attention. The main OT tool on these spaces is the Wasserstein distance which suffers from a heavy computational burden. On Euclidean spaces, a popular alternative is the Sliced-Wasserstein distance, which leverages a closed-form solution of the Wasserstein distance in one dimension, but which is not readily available on manifolds. In this work, we derive general constructions of Sliced-Wasserstein distances on Cartan-Hadamard manifolds, Riemannian manifolds with non-positive curvature, which include among others Hyperbolic spaces or the space of Symmetric Positive Definite matrices. Then, we propose different applications. Additionally, we derive non-parametric schemes to minimize these new distances by approximating their Wasserstein gradient flows.